def get_xy_ref(self):
'''
Gets x in function of 'xi' and 'eta'
Gets y in function of 'xi; and 'eta'
X is the Sum of N_i * X_i for each node of the element.
Y is the Sum of N_i * Y_i for each node of the element.
'''
if self.Ne_ref is None:
self.get_Ne_ref()
x_coord = 0
y_coord = 0
for i in range(self.num_nodes):
x_coord = x_coord + (self.nodes[i].x * self.Ne_ref[i])
y_coord = y_coord + (self.nodes[i].y * self.Ne_ref[i])
for i in sy.preorder_traversal(x_coord):
if isinstance(i, sy.Float) and abs(i) < 1e-10:
x_coord = x_coord.subs(i, round(i, 1))
elif isinstance(i, sy.Float):
x_coord = x_coord.subs(i, round(i, 15))
for i in sy.preorder_traversal(y_coord):
if isinstance(i, sy.Float) and abs(i) < 1e-10:
y_coord = y_coord.subs(i, round(i, 1))
elif isinstance(i, sy.Float):
y_coord = y_coord.subs(i, round(i, 15))
self.x_coord = x_coord
self.y_coord = y_coord
def checkanswer(self, user_answer):
user_answer = user_answer.lower()
user_answer = user_answer.replace('^', '**')
user_answer = parse_expr(user_answer, transformations=transformations, evaluate=False)
print('user stuff', user_answer, type(user_answer))
if isinstance(user_answer, sy.Pow):
if user_answer.args[1] == -1:
# print('My fault!')
print(user_answer.args)
if isinstance(user_answer.args[0], sy.Pow):
user_base = user_answer.args[0].args[0]
user_expo = -user_answer.args[0].args[1]
user_answer = sy.Pow(user_base, sy.simplify(user_expo), evaluate=False)
print('alt', user_answer, type(user_answer))
for arg in sy.preorder_traversal(user_answer):
print(arg)
else:
user_base = user_answer.args[0]
user_expo = user_answer.args[1]
user_answer = sy.Pow(user_base, sy.simplify(user_expo), evaluate=False)
print('standard', user_answer, type(user_answer))
for arg in sy.preorder_traversal(user_answer):
print(arg)
print('self.answer', self.answer)
answer = self.answer
# answer = parse_expr(str(self.answer), transformations=transformations)
print('answer stuff', answer, type(answer))
for arg in sy.preorder_traversal(answer):
print(arg)
return user_answer == answer
def cse_postprocess(cse_output):
replaced, reduced = cse_output
i = 0
while i < len(replaced):
sym, expr = replaced[i]
# Search through replaced expressions for negative symbols
if (expr.func == sp.Mul and len(expr.args) == 2 and \
any((arg.func == sp.Symbol) for arg in expr.args) and \
any((arg == sp.S.NegativeOne or '_NegativeOne_' in str(arg)) for arg in expr.args)):
for k in range(i + 1, len(replaced)):
if sym in replaced[k][1].free_symbols:
replaced[k] = (replaced[k][0],
replaced[k][1].subs(sym, expr))
for k in range(len(reduced)):
if sym in reduced[k].free_symbols:
reduced[k] = reduced[k].subs(sym, expr)
# Remove the replaced expression from the list
replaced.pop(i)
if i != 0: i -= 1
# Search through replaced expressions for addition/product of 2 or less symbols
if ((expr.func == sp.Add or expr.func == sp.Mul) and 0 < len(expr.args) < 3 and \
all((arg.func == sp.Symbol or arg.is_integer or arg.is_rational) for arg in expr.args)) or \
(expr.func == sp.Pow and expr.args[0].func == sp.Symbol and expr.args[1] == 2):
sym_count = 0 # Count the number of occurrences of the substituted symbol
for k in range(len(replaced) - i):
# Check if the substituted symbol appears in the replaced expressions
if sym in replaced[i + k][1].free_symbols:
for arg in sp.preorder_traversal(replaced[i + k][1]):
if arg.func == sp.Symbol and str(arg) == str(sym):
sym_count += 1
for k in range(len(reduced)):
# Check if the substituted symbol appears in the reduced expression
if sym in reduced[k].free_symbols:
for arg in sp.preorder_traversal(reduced[k]):
if arg.func == sp.Symbol and str(arg) == str(sym):
sym_count += 1
# If the number of occurrences of the substituted symbol is 2 or less, back-substitute
if 0 < sym_count < 3:
for k in range(i + 1, len(replaced)):
if sym in replaced[k][1].free_symbols:
replaced[k] = (replaced[k][0],
replaced[k][1].subs(sym, expr))
for k in range(len(reduced)):
if sym in reduced[k].free_symbols:
reduced[k] = reduced[k].subs(sym, expr)
# Remove the replaced expression from the list
replaced.pop(i)
i -= 1
i += 1
return replaced, reduced
def get_index_derivatives(expr):
"""
"""
coord = ['x', 'y', 'z']
d = OrderedDict()
for c in coord:
d[c] = 0
ops = [
a for a in preorder_traversal(expr)
if isinstance(a, _partial_derivatives)
]
for i in ops:
op = type(i)
if isinstance(i, dx):
d['x'] += 1
elif isinstance(i, dy):
d['y'] += 1
elif isinstance(i, dz):
d['z'] += 1
return d
def get_index_logical_derivatives(expr):
"""
"""
coord = ['x1', 'x2', 'x3']
d = OrderedDict()
for c in coord:
d[c] = 0
ops = [
a for a in preorder_traversal(expr)
if isinstance(a, _logical_partial_derivatives)
]
for i in ops:
op = type(i)
if isinstance(i, dx1):
d['x1'] += 1
elif isinstance(i, dx2):
d['x2'] += 1
elif isinstance(i, dx3):
d['x3'] += 1
return d
def gelatize(expr, dim):
# ... in the case of a Lambda expression
args = None
if isinstance(expr, Lambda):
args = expr.variables
expr = expr.expr
# ...
# ... we first need to find the ordered list of generic operators
ops = [a for a in preorder_traversal(expr) if isinstance(a, _generic_ops)]
# ...
# ...
for i in ops:
# if i = Grad(u) then type(i) is Grad
op = type(i)
new = eval('{0}_{1}d'.format(op, dim))
expr = expr.subs(op, new)
# ...
if args:
return Lambda(args, expr)
else:
return expr
def filter_expr2(expr, threshold=0.01):
"""Sets all constants under threshold to 0
TODO: Test"""
for a in sym.preorder_traversal(expr):
if isinstance(a, sym.Float) and a < threshold:
expr = expr.subs(a, 0)
return expr
def _eval_is_zero(self):
if self.shape:
return
if len(self.args) == 2:
if self.is_extended_real:
if self.min().is_extended_positive:
return False
if self.max().is_extended_negative:
return False
from sympy import preorder_traversal, KroneckerDelta
delta = None
for arg in preorder_traversal(self):
if isinstance(arg, KroneckerDelta):
delta = arg
break
if delta is not None:
#to prevent infinite loop!
this = self._subs(delta, S.Zero)
if this is self:
return
delta0 = this.is_zero
this = self._subs(delta, S.One)
if this is self:
return
delta1 = this.is_zero
if delta0 and delta1:
return True
if delta0 == False and delta1 == False:
return False
def round_sympy_expr(expr, decimals):
"""Returns the expression with every float rounded to the given number of decimals."""
rounded_expr = expr
for a in sympy.preorder_traversal(expr):
if isinstance(a, sympy.Float):
rounded_expr = rounded_expr.subs(a, round(a, decimals))
return rounded_expr
def traversal(expr):
"""Traverse sympy expression tree
Args:
expr (sympy expression)
"""
return list(preorder_traversal(expr))
def round_sympy_expr(expr, decimals):
"""Returns the expression with every float rounded to the given number of decimals."""
rounded_expr = expr
for a in sympy.preorder_traversal(expr):
if isinstance(a, sympy.Float):
rounded_expr = rounded_expr.subs(a, round(a, decimals))
return rounded_expr
def expr_to_infix(expr, mul_add_arity_fixed=False):
"""
Returns preorder traversal of the symbolic expression
"""
pre = []
pre_arity = []
for expr_node in snp.preorder_traversal(expr):
if expr_node.func.is_symbol:
pre.append(expr_node.name)
pre_arity.append(len(expr_node.args))
elif expr_node.is_Function or expr_node.is_Add or expr_node.is_Mul or expr_node.is_Pow:
if mul_add_arity_fixed and (expr_node.is_Add or expr_node.is_Mul):
for i in range(len(expr_node.args) - 1):
pre_arity.append(2)
pre.append(type(expr_node).__name__)
else:
pre_arity.append(len(expr_node.args))
pre.append(type(expr_node).__name__)
elif expr_node.is_constant():
pre.append(float(expr_node))
post_arity.append(len(expr_node.args))
return pre, pre_arity
def _extract_advection(self, order_dict):
node = Add(*flatten(order_dict.itervalues()))
divs = filter(lambda x: isinstance(x, div), preorder_traversal(node))
div_args = map(lambda d: d.args[0], divs)
split_div_args = flatten(map(self._split_on_add, div_args))
div_args = filter(lambda n: not self._has_grad(n), split_div_args)
return Add(*div_args)
def traversal(expr):
'''Traverse sympy expression tree
Args:
expr (sympy expression)
'''
return list(preorder_traversal(expr))
def to_monomials(expr):
try:
e = expr.expand()
except Exception:
if hasattr(expr, 'copy'):
e = expr.copy()
else:
e = expr
while True:
has_change = False
for x in preorder_traversal(e):
# print("Doing %s" % str(x))
if hasattr(x, 'is_Mul') and x.is_Mul:
# print("here")
for i, y in enumerate(x.args):
if hasattr(y, 'is_Add') and y.is_Add and\
noncommutative(y.args):
if i == 0:
terms = add_list_head(y.args, x.args[1:])
elif i == len(y.args)-1:
terms = add_list_tail(y.args, x.args[:-1])
else:
terms = add_list_middle(
y.args, x.args[:i], x.args[i+1:])
enew = e.xreplace({x: terms})
if enew != e:
has_change = True
e = enew
break
if has_change:
break
if not has_change:
break
return e
def _extract_advection(self, order_dict):
node = Add(*flatten(order_dict.itervalues()))
divs = filter(lambda x: isinstance(x, div), preorder_traversal(node))
div_args = map(lambda d: d.args[0], divs)
split_div_args = flatten(map(self._split_on_add, div_args))
div_args = filter(lambda n: not self._has_grad(n), split_div_args)
return Add(*div_args)
def _conditional_replace(expr, condition, replacement):
for x in preorder_traversal(expr):
try:
if condition(x):
expr = expr.xreplace({x: replacement(x)})
except AttributeError: # scalar ops like Add won't have is_Trace
pass
return expr
def contain_exp(expr):
""" Return True if expression contains the exponential operator.
"""
for arg in sym.preorder_traversal(expr):
# check if there is an exponential function
if arg.func == sym.exp:
return True
return False
def _extract_potential(self, order_dict):
second_order = order_dict.get(2, 0)
div_grads = filter(self._is_div_grad, preorder_traversal(second_order))
div_grad_args = map(lambda d: d.args[0], div_grads)
potentials = flatten((map(self._find_grad_args, div_grad_args)))
if len(potentials) == 1:
potentials = potentials[0]
return potentials
def tree_size(eq):
i = 0
for e in sympy.preorder_traversal(self.expr):
if e.is_Integer:
i += int(abs(e))
continue
i += 1
return i
def symbolic_equal(expression1, expression2, x, mu):
locals = {"x": x, "mu": mu}
difference = sympify(expression1, locals=locals) - sympify(expression2, locals=locals)
difference = simplify(difference)
for node in preorder_traversal(difference):
if isinstance(node, Float):
difference = difference.subs(node, round(node, 10))
return difference == 0
def tree_size(eq): i = 0 for e in sympy.preorder_traversal(self.expr): if e.is_Integer: i += int(abs(e)) continue i+=1 return i
def _extract_potential(self, order_dict):
second_order = order_dict.get(2, 0)
div_grads = filter(self._is_div_grad, preorder_traversal(second_order))
div_grad_args = map(lambda d: d.args[0], div_grads)
potentials = flatten((map(self._find_grad_args, div_grad_args)))
if len(potentials) == 1:
potentials = potentials[0]
return potentials
def count_occurrences2(expr):
"""
Count atom occurrences in an expression.
"""
result = {}
for sub_expr in sp.preorder_traversal(expr):
if sub_expr.is_Atom:
result[sub_expr] = result.get(sub_expr, 0) + 1
return result
def exprIsPositive(expr):
'''This function is *incredibly* sketchy. It sometimes determines
whether an expression is positive or negative. It's true for "x*y"
and false for "-x*y".'''
for a in preorder_traversal(expr):
if issubclass(a.func, Number):
if a < 0:
return False
def generate_propagator_solver(self, output_timestep_symbol="__h"):
r"""
Generate the propagator matrix and symbolic expressions for propagator-based updates; return as JSON.
"""
#
# generate the propagator matrix
#
P = sympy.simplify(sympy.exp(self.A_ * sympy.Symbol(output_timestep_symbol)))
if sympy.I in sympy.preorder_traversal(P):
raise PropagatorGenerationException("The imaginary unit was found in the propagator matrix. This can happen if the dynamical system that was passed to ode-toolbox is unstable, i.e. one or more state variables will diverge to minus or positive infinity.")
#
# generate symbols for each nonzero entry of the propagator matrix
#
P_sym = sympy.zeros(*P.shape) # each entry in the propagator matrix is assigned its own symbol
P_expr = {} # the expression corresponding to each propagator symbol
update_expr = {} # keys are str(variable symbol), values are str(expressions) that evaluate to the new value of the corresponding key
for row in range(P_sym.shape[0]):
if not _is_zero(self.c_[row]):
raise PropagatorGenerationException("For symbol " + str(self.x_[row]) + ": nonlinear part should be zero for propagators")
if not _is_zero(self.b_[row]) and self.shape_order_from_system_matrix(row) > 1:
raise PropagatorGenerationException("For symbol " + str(self.x_[row]) + ": higher-order inhomogeneous ODEs are not supported")
update_expr_terms = []
for col in range(P_sym.shape[1]):
if not _is_zero(P[row, col]):
sym_str = "__P__{}__{}".format(str(self.x_[row]), str(self.x_[col]))
P_sym[row, col] = sympy.parsing.sympy_parser.parse_expr(sym_str, global_dict=Shape._sympy_globals)
P_expr[sym_str] = P[row, col]
if _is_zero(self.b_[row]):
# homogeneous ODE
update_expr_terms.append(sym_str + " * " + str(self.x_[col]))
else:
# inhomogeneous ODE
particular_solution = -self.b_[row] / self.A_[row, row]
update_expr_terms.append(sym_str + " * (" + str(self.x_[col]) + " - (" + str(particular_solution) + "))" + " + (" + str(particular_solution) + ")")
update_expr[str(self.x_[row])] = " + ".join(update_expr_terms)
update_expr[str(self.x_[row])] = sympy.parsing.sympy_parser.parse_expr(update_expr[str(self.x_[row])], global_dict=Shape._sympy_globals)
if not _is_zero(self.b_[row]):
# only simplify in case an inhomogeneous term is present
update_expr[str(self.x_[row])] = sympy.simplify(update_expr[str(self.x_[row])])
all_state_symbols = [str(sym) for sym in self.x_]
initial_values = {sym: str(self.get_initial_value(sym)) for sym in all_state_symbols}
solver_dict = {"propagators": P_expr,
"update_expressions": update_expr,
"state_variables": all_state_symbols,
"initial_values": initial_values}
return solver_dict
def exprIsPositive(expr):
'''This function is *incredibly* sketchy. It sometimes determines
whether an expression is positive or negative. It's true for "x*y"
and false for "-x*y".'''
for a in preorder_traversal(expr):
if issubclass(a.func,Number):
if a<0:
return False
def list_indices(expr):
"Gathers the list of pulse indices present in expr. Returns a sorted list"
return sorted(
list(
set(
chain(*[
map(abs, a.k) for a in sy.preorder_traversal(expr)
if isinstance(a, Signal)
]))))
def round_nested(expression: sp.Expr, n_decimals: int) -> sp.Expr:
for node in sp.preorder_traversal(expression):
if node.free_symbols:
continue
if isinstance(node, (float, sp.Float)):
expression = expression.subs(node, round(node, n_decimals))
if isinstance(node, sp.Pow) and node.args[1] == 1 / 2:
expression = expression.subs(node, round(node.n(), n_decimals))
return expression
def _is_div_grad(node):
"""Returns true node is a div with a grad argument inside"""
is_div = isinstance(node, div)
div_grad = False
if is_div:
for arg in preorder_traversal(node.args[0]):
if isinstance(arg, grad):
div_grad = True
break
return div_grad
def _is_div_grad(node):
"""Returns true node is a div with a grad argument inside"""
is_div = isinstance(node, div)
div_grad = False
if is_div:
for arg in preorder_traversal(node.args[0]):
if isinstance(arg, grad):
div_grad = True
break
return div_grad
def round_and_simplify(stuff):
simplified_stuff = simplify(stuff)
rounded_stuff = simplified_stuff
for a in preorder_traversal(rounded_stuff):
if isinstance(a, Float):
rounded_stuff = rounded_stuff.subs(a, round(a, 10))
rounded_and_simplified_stuff = simplify(rounded_stuff)
return rounded_and_simplified_stuff
def round_and_simplify(stuff):
simplified_stuff = simplify(stuff)
rounded_stuff = simplified_stuff
for a in preorder_traversal(rounded_stuff):
if isinstance(a, Float):
rounded_stuff = rounded_stuff.subs(a, round(a, 10))
rounded_and_simplified_stuff = simplify(rounded_stuff)
return rounded_and_simplified_stuff
def extract_funcs(expr):
"""
Given a sympy expression, return a tuple of all the functions contained
within that expression.
"""
functions = set()
for arg in preorder_traversal(expr):
if isinstance(arg, Function):
functions.add(arg)
return tuple(functions)
def calc_tree_size(self):
i = 0
p = 0
for e in sympy.preorder_traversal(self.expr):
if e.is_Integer:
i += int(abs(e))
continue
if e.is_Function:
p += 2 # add an extra penalty
i += 1
return i, i + p
def dse_dimensions(expr):
"""
Collect all function dimensions used in a sympy expression.
"""
dimensions = []
for e in preorder_traversal(expr):
if isinstance(e, SymbolicData):
dimensions += [i for i in e.indices if i not in dimensions]
return dimensions
def calc_jac_size(self):
i, p = 0, 0
for jac in self.jac:
for e in sympy.preorder_traversal(jac):
if e.is_Integer:
i += int(abs(e))
continue
if e.is_Function:
p += 2 # add an extra penalty
i += 1
return i, i + p
def calc_tree_size(self): i = 0 p = 0 for e in sympy.preorder_traversal(self.expr): if e.is_Integer: i += int(abs(e)) continue if e.is_Function: p += 2 # add an extra penalty i+=1 return i, i+p
def calc_jac_size(self): i,p = 0,0 for jac in self.jac: for e in sympy.preorder_traversal(jac): if e.is_Integer: i += int(abs(e)) continue if e.is_Function: p += 2 # add an extra penalty i+=1 return i, i+p
def indexed_terms_appearing_in(term, indexed, only_subscripts=False, do_traversal=False):
indexed_terms_set = set()
for subterm in preorder_traversal(term) if do_traversal else flatten(term.args, cls=Add):
try:
with bind_Mul_indexed(subterm, indexed) as (_, subscripts):
indexed_terms_set.add(subscripts if only_subscripts else indexed[subscripts])
except DestructuringError:
continue
return list(indexed_terms_set)
def encode(self,expr): # print expr iis = [] ffs = [] for i,e in enumerate(sympy.preorder_traversal(expr)): # print i, e.func ii, ff = self.mapper.get(e) iis = iis + ii if ff is not None: iis.append(len(ffs)) ffs = ffs + ff # print " ", ii, ff return iis, ffs
def symbols_from_expr(expr, include_numbers=False, include_derivatives=False):
"""
Returns a set of all symbols of an expression
Arguments:
----------
expr : sympy expression
A sympy expression containing sympy.Symbols or sympy.AppliedUndef
functions.
include_numbers : bool
If True numbers will also be returned
include_derivatives : bool
If True derivatives will be returned instead of its variables
"""
from sympy.core.function import AppliedUndef
symbols = set()
pt = sp.preorder_traversal(expr)
for node in pt:
# Do not traverse AppliedUndef
if isinstance(node, AppliedUndef):
pt.skip()
symbols.add(node)
elif isinstance(node, sp.Symbol) and not isinstance(node, sp.Dummy) \
and node.name:
symbols.add(node)
elif include_numbers and isinstance(node, sp.Number):
symbols.add(node)
elif include_derivatives and isinstance(node, sp.Derivative):
# Do not traverse Derivative
pt.skip()
symbols.add(node)
return symbols
def rearangeForDerivs(self,eoms,secondOrder=False):
suffix = self.derivative_suffix
if secondOrder:
suffix += suffix
derivSet = set()
for eom in eoms:
for x in sympy.preorder_traversal(eom):
if type(x) == sympy.Symbol and suffix in str(x):
derivSet.add(x)
derivList = list(derivSet)
if len(derivList) == 0:
raise Exception("No time derivative was found in eom: '{0}', where derivative has suffix of '{1}'".format(eom,suffix))
solns = sympy.solve(eoms,derivList)
if len(solns) < len(derivList):
raise Exception("Too few solutions, {2}, were found for '{1}'".format(eom,derivList,solns))
if len(solns) > len(derivList):
raise Exception("Too many solutions, {2}, were found for '{1}'".format(eom,derivList,solns))
for deriv in derivList:
try:
solns[deriv]
except KeyError:
raise Exception("Couldn't find solution for {1}, in solutions {2}, in EOMs '{0}'".format(eom,deriv,solns))
return solns
def positif(expression, variable=None, strict=False, _niveau=0, _changement_variable=None):
"""Retourne l'ensemble sur lequel une expression à variable réelle est positive (resp. strictement positive)."""
from .sympy_functions import factor
# L'étude du signe se fait dans R, on indique donc à sympy que la variable est réelle.
if variable is None:
variable = extract_var(expression)
if hasattr(expression, "subs"):
old_variable = variable
variable = Dummy(real=True)
expression = expression.subs({old_variable:variable})
# Ensemble de définition et périodicité
ens_def = ensemble_definition(expression, variable)
T = periode(expression, variable)
if T not in (0, oo):
ens_def &= Intervalle(0, T)
if param.debug:
print('Fonction périodique %s détectée. Résolution sur [0;%s]' % (expression, T))
# On remplace sqrt(x^2) par |x|.
a = Wild('a', exclude=[expression])
dic = expression.match(sqrt(a**2))
if dic is not None:
sub_expr = dic[a]
# On évite de détecter sqrt(x) comme étant sqrt(sqrt(x)**2) !
if not (sub_expr.is_Pow and sub_expr.as_base_exp()[1].is_integer):
expression = expression.replace(sqrt(a**2),Abs(a))
expression = expression.replace(Abs(sqrt(a)),sqrt(a))
del a
# On factorise au maximum.
try:
expression = factor(expression, variable, "R", decomposer_entiers = False)
except NotImplementedError:
if param.debug:
print("Warning: Factorisation impossible de ", expression)
if expression.is_Pow and expression.as_base_exp()[1].is_rational:
base, p = expression.as_base_exp()
# Le dénominateur ne doit pas s'annuler :
if p < 0:
strict = True
if p.is_integer and p.is_even:
if strict:
return ens_def & (R - (positif(base, variable, strict=False) - positif(base, variable, strict=True)))
else:
return ens_def
else:
return ens_def & positif(base, variable, strict=strict)
if expression.is_Mul:
posit = R
posit_nul = R
for facteur in expression.args:
# pos : ensemble des valeurs pour lequelles l'expression est positive
# pos_nul : ensemble des valeurs pour lequelles l'expression est positive ou nulle
pos = positif(facteur, variable, strict = True)
pos_nul = positif(facteur, variable, strict = False)
# posit : les deux sont strictements positifs, ou les deux sont strictements négatifs
# posit_nul : les deux sont positifs ou nuls, ou les deux sont négatifs ou nuls
posit, posit_nul = ((posit & pos) + (-posit_nul & -pos_nul) & ens_def,
((posit_nul & pos_nul) + (-posit & -pos)) & ens_def)
if strict:
return posit
else:
return posit_nul
if expression.is_positive is True: # > 0
return ens_def
if expression.is_negative is True: # < 0
return vide
if expression.is_positive is False and strict: # <= 0
return vide
if expression.is_negative is False and not strict: # >= 0
return ens_def
if expression.is_zero is True and not strict: # == 0
return ens_def
if isinstance(expression, (int, float)):
if expression > 0 or (expression == 0 and not strict):
return ens_def
else:
return vide
# Inutile de se préoccuper de l'ensemble de définition pour les fonctions polynômiales.
if expression.is_polynomial():
P = expression.as_poly(variable)
if P.degree() == 1:
a, b = P.all_coeffs()
if a > 0:
return Intervalle(-b/a, +oo, inf_inclus = not strict)
else: # a<0 (car a != 0)
return Intervalle(-oo, -b/a, sup_inclus = not strict)
elif P.degree() == 2:
a, b, c = P.all_coeffs()
d = b**2 - 4*a*c
if d > 0:
x1 = (-b - sqrt(d))/(2*a)
x2 = (-b + sqrt(d))/(2*a)
x1, x2 = min(x1, x2), max(x1, x2)
if a > 0:
return Intervalle(-oo, x1, sup_inclus = not strict) + Intervalle(x2, +oo, inf_inclus = not strict)
else: # a<0 (car a != 0)
return Intervalle(x1, x2, inf_inclus = not strict, sup_inclus = not strict)
elif d == 0:
x0 = -b/(2*a)
if a > 0:
return Intervalle(-oo, x0, sup_inclus = not strict) + Intervalle(x0, +oo, inf_inclus = not strict)
else:
return Intervalle(x0, x0, sup_inclus = not strict)
else: # d < 0
if a > 0:
return R
else:
return vide
# Valeur absolue seule:
if isinstance(expression, Abs):
return ens_def
# Valeur absolue: cas général. L'idée est simplement de supprimer la valeur
# absolue en faisant deux cas: |f(x)| = f(x) si f(x)>=0 et -f(x) si f(x)<0.
# (Le `for` est trompeur: on ne supprime qu'une seule valeur absolue à la fois, la récursivité fait le reste...)
for subexpr in expression.find(Abs):
# subexpr = |f(x)|
# val = f(x)
val = subexpr.args[0]
pos = positif(val, variable)
# g(|f(x)|, x) > 0 <=> g(f(x), x) > 0 et f(x) >= 0 ou g(-f(x), x) > 0 et f(x) < 0
return ((positif(expression.xreplace({subexpr: val}), variable, strict=strict) & pos) |
(positif(expression.xreplace({subexpr: -val}), variable, strict=strict) & -pos))
# Puissances
# Résolution de a*x^q-b > 0, où q n'est pas entier
if expression.is_Add:
a = Wild('a', exclude=[variable, 0])
q = Wild('q', exclude=[variable, 1])
b = Wild('b', exclude=[variable])
X = Wild('X')
# match ne semble pas fonctionner avec a*X**q - b
match = expression.match(a*X**q + b)
if match and X in match:
a = match[a]
q = match[q]
b = -match[b]/a
X = match[X]
# Le cas où q est entier est déjà traité par ailleurs (polynômes).
# Lorsque q est de la forme 1/n, on peut définir la fonction x−>x^(1/n)
# sur R si n est impair et sur R+ sinon.
# Dans tous les autres cas, on définit la fonction x->x^q sur ]0;+oo[.
if not q.is_integer:
if a.is_negative:
# si a < 0, a*x^q-b > 0 <=> x^q-b/a < 0 <=> non(x^q-b/a >= 0)
strict = not strict
if b.is_negative:
# pour X > 0, X^(2/3)-b > X^2/3 > 0
sols = ens_def
else:
# pour X > 0, X^(2/3)-b > 0 <=> X > b^(3/2)
sols = ens_def & positif(X - b**(1/q), strict=strict)
# si a < 0, a*x^q-b > 0 <=> x^q-b/a < 0 <=> non(x^q-b/a >= 0)
if a.is_negative:
sols = ens_def - sols
return sols
del a, q, b, X
# résolution de a*sqrt(B)-c*sqrt(D) > 0 où signe(c) = signe(a)
if expression.is_Add:
a = Wild('a', exclude=[0, variable])
B = Wild('B', exclude=[0])
c = Wild('c', exclude=[0, variable])
D = Wild('D', exclude=[0])
match = expression.match(a*sqrt(B) - c*sqrt(D))
if match:
a = match[a]
B = match[B]
c = match[c]
D = match[D]
if a.is_positive and c.is_positive:
return ens_def & positif(a**2*B - c**2*D, variable, strict=strict)
elif a.is_negative and c.is_negative:
return ens_def & positif(c**2*D - a**2*B, variable, strict=strict)
del a, B, c, D
# Logarithme :
if isinstance(expression, ln):
return positif(expression.args[0] - 1, variable, strict=strict)
# Sommes contenant des logarithmes :
# Résolution de ln(X1) + ln(X2) + ... + b > 0, où X1=f1(x), X2 = f2(x) ...
if expression.is_Add and expression.has(ln):
args = expression.args
liste_constantes = []
liste_ln = []
# Premier passage : on remplace a*ln(X1) par ln(X1**a)
for arg in args:
if getattr(arg, "is_Mul", False):
liste_constantes = []
liste_ln = []
for facteur in arg.args:
if isinstance(facteur, ln) and is_var(facteur, variable):
liste_ln.append(facteur)
elif not is_var(facteur, variable):
liste_constantes.append(facteur)
if len(liste_constantes) == len(arg.args) - 1 and len(liste_ln) == 1:
expression += ln(liste_ln[0].args[0]**Add(*liste_constantes)) - arg
# Deuxième passage : ln(X1)+ln(X2)+b>0 <=> X1*X2-exp(-b)>0
for arg in args:
if isinstance(arg, ln) and hasattr(arg, "has") and arg.has(variable):
liste_ln.append(arg)
elif not hasattr(arg, "has") or not arg.has(variable):
liste_constantes.append(arg)
if liste_ln and len(liste_ln) + len(liste_constantes) == len(args):
# ln(X1)+ln(X2)+b>0 <=> X1*X2-exp(-b)>0
contenu_log = Mul(*(logarithme.args[0] for logarithme in liste_ln))
contenu_cst = exp(- Add(*liste_constantes))
return ens_def & positif(contenu_log - contenu_cst, variable, strict=strict)
# Cas très particulier : on utilise le fait que ln(x)<=x-1 sur ]0;+oo[
expr = expression
changements = False
for arg in expr.args:
if isinstance(arg, ln):
changements = True
expr += arg.args[0] + 1 - arg
if changements:
try:
non_positif = positif(-expr, variable, strict = not strict) # complementaire
(ens_def - positif(expr, variable, strict = not strict) == vide)
if (ens_def - non_positif == vide):
return vide
except NotImplementedError:
pass
# Si aucune autre méthode n'a fonctionné, on tente ln(a)+ln(b)>0 <=> a*b>1 (pour a>0 et b>0)
expr = Mul(*(exp(arg) for arg in expression.args)) - 1
try:
return ens_def*positif(expr, variable, strict=strict)
except NotImplementedError:
pass
# Exponentielle
# Résolution de a*exp(f(x)) + b > 0
if expression.is_Add and expression.has(exp):
a = Wild('a', exclude=[variable, 0])
b = Wild('b', exclude=[variable])
X = Wild('X')
match = expression.match(a*exp(X) + b)
if match:
a = match[a]
b = match[b]
X = match[X]
if is_pos(b):
if is_pos(a):
return ens_def
elif is_neg(a):
# l'ensemble de définition ne change pas
return positif(- X + ln(-b/a), variable, strict=strict)
elif is_neg(b):
if is_pos(a):
return positif(X - ln(-b/a), variable, strict=strict)
elif is_neg(a):
return vide
del a, b, X
# Cas très particulier : on utilise le fait que exp(x)>=x+1 sur R
expr = expression
changements = False
for arg in expr.args:
if isinstance(arg, exp):
changements = True
expr += arg.args[0] + 1 - arg
if changements and (ens_def - positif(expr, variable, strict=strict) == vide):
return ens_def
# Cas très particulier : si a≥0, b*(a*x-1+exp(x))>0 <=> b*x>0
a = Wild('a', exclude=[variable, 0])
b = Wild('b', exclude=[variable, 0])
X = Wild('X')
match = expression.match(b*(a*X - 1 + exp(X)))
if match is not None and X in match:
a = match[a]
if a >= 0:
b = match[b]
X = match[X]
return positif(b*X, variable, strict=strict)
del a, b, X
# Aucun cas connu n'a été détecté, on tente un changement de variable.
# NB: _niveau sert à éviter les récursions infinies !
if _niveau > 10:
print("Infinite recursion suspected. Aborting...")
raise NotImplementedError
# La technique globalement est la suivante :
# sqrt(x² + 5) - 9 > 0 <=> sqrt(X) - 9 > 0 (avec X = x² + 5) <=> X in ]3;+oo[ <=> X - 3 > 0 <=> x² + 5 - 3 > 0, etc.
def _resolution_par_substitution(expr, X):
try:
solutions_intermediaires = positif(expr, X, strict=strict,
_niveau=_niveau + 1,
_changement_variable=Lambda(variable,sub))
solution = vide
for intervalle in solutions_intermediaires.intervalles:
sol = R
a = intervalle.inf
b = intervalle.sup
if a != - oo:
sol &= positif(sub - a, variable, strict=(not intervalle.inf_inclus))
if b != oo:
sol &= positif(b - sub, variable, strict=(not intervalle.sup_inclus))
solution += sol
return ens_def & solution
except NotImplementedError:
return None
# On commence par lister tous les changements de variable potentiels,
# c'est-à-dire toutes les sous-expressions qui contiennent la variable.
changements = sorted(set(subexpr for subexpr in preorder_traversal(expression)
if subexpr.has(variable)) - {variable, expression},
key=count_ops)
a = Wild('a', exclude=[variable, 0])
b = Wild('b', exclude=[variable, 0])
X = Dummy(real=True)
changements_incomplets = []
for sub in changements:
if _changement_variable is not None:
enchainement = _changement_variable(sub)
if enchainement.match(a*variable) or enchainement.match(a*abs(variable)):
# On enchaîne deux changements de variable réciproque,
# comme ln() et exp() ou encore x->x² et x-> sqrt(x).
# Ce qui conduirait à une récursion infinie.
continue
match = expression.match(a*sub + b)
if match and match[a] in (-1, 1):
# Ceci conduirait à des récursions infinies du genre :
# sqrt(x² + 5) - 9 > 0 <=> X - 9 > 0 (avec X = sqrt(x² + 5)) <=> X in ]9;+oo[ <=> X - 9 > 0 <=> sqrt(x² + 5) - 9 > 0, etc.
continue
# On tente le changement de variable.
new = expression.subs(sub, X)
if new.has(variable):
# Changement de variable incomplet.
# On le garde en réserve pour une 2e passe si on n'a pas trouvé mieux.
changements_incomplets.append(sub)
else:
sols = _resolution_par_substitution(new, X)
if sols is not None:
return sols
for sub in changements_incomplets:
# On cherche si le changement de variable est inversible.
# Si oui, au lieu de remplacer f(x) par X, on remplace x par f-¹(X)
# Bizarrement, l'assertion real=True fait planter le solver() de sympy
# lorsqu'on lance solve(sqrt(x)-y, x). On utilise donc des variables
# temporaires sans assertions.
var1 = Dummy()
var2 = Dummy()
antecedents = solve((sub - X).xreplace({variable:var1, X:var2}), var1)
if len(antecedents) == 1:
new = expression.subs(variable, antecedents[0].xreplace({var1:variable, var2:X}))
sols = _resolution_par_substitution(new, X)
if sols is not None:
return sols
# En dernier ressort, résolution par continuité.
# Les fonctions usuelles sont continues sur tout intervalle de leur ensemble de définition.
# Il suffit donc de rechercher les zéros de la fonction, et d'évaluer le signe
# de l'expression sur chaque tronçon.
# Cette méthode est simple, mais elle suppose que **tous** les zéros aient bien
# été trouvés par solve(), ce qui n'est pas toujours le cas.
# C'est pourquoi elle est proposée seulement en dernier recours.
if param.debug:
print("Warning: résolution par continuité. Les résultats peuvent être\n"
"faux si certaines racines ne sont pas touvées !")
racines = nul(expression, variable, intervalle=False)
solutions = vide
for intervalle in ens_def.intervalles:
inf = intervalle.inf
sup = intervalle.sup
decoupage = {S(inf), S(sup)}
for borne in decoupage:
if borne in intervalle:
if expression.subs(variable, borne) > 0:
solutions |= Intervalle(borne, borne)
for rac in racines:
if rac in intervalle:
if not strict:
solutions |= Intervalle(rac, rac)
decoupage.add(rac)
decoupage = sorted(decoupage)
for val1, val2 in zip(decoupage[:-1], decoupage[1:]):
if val1 == -oo and val2 == oo:
milieu = 0
elif val2 == oo:
milieu = val1 + 1
elif val1 == -oo:
milieu = val2 - 1
else:
milieu = (val1 + val2)/2
if expression.subs(variable, milieu) > 0:
solutions |= Intervalle(val1, val2, False, False)
return ens_def&solutions
def _has_grad(node):
"""Return true if node has a grad in it"""
return len(filter(lambda n: isinstance(n, grad),
preorder_traversal(node))) > 0
def _has_div(node):
"""Return true if node has a div in it"""
return len(filter(lambda n: isinstance(n, div),
preorder_traversal(node))) > 0
def _extract_diffusion(self, order_dict):
second_order = order_dict.get(2, 0)
div_grads = filter(self._is_div_grad, preorder_traversal(second_order))
div_grad_args = map(lambda d: d.args[0], div_grads)
grads = map(self._find_grad_coefficient, div_grad_args)
return Add(*list(grads))
def has_sqrt(sympy_obj):
return any(el.func is C.Pow and el.args[-1] is S.Half
for el in preorder_traversal(sympy_obj))